Applications of the Integral

Average Value of a Function

Length of a Graph

We could also have observed that in a
previous section
we found the
area under this function to be 4, which is by definition the product
of
2Π
and the average value.

Problem :
Suppose Eleanor invests 100 dollars in an account that is compounded continuously
with an annual yield of 5 percent, so that the number of dollars in the account after
t
years is given by
A(t) = 100(1.05)t
. What is the average amount of money in her
account over the first 3 years?

We have

=

100(1.05)tdt

=

|03

=

(1.053 - 1)

or approximately 107.69
dollars.

Problem :
What is the average
y
-coordinate of a point on the upper half of the unit circle centered
at the origin? (You may use that
dx = (x/2) + (1/2)sin-1(x)
.)

The upper half of the unit circle centered at the origin is the graph of the function
f (x) =
on the interval
[- 1, 1]
. The average value of
f
on this interval
equals

dx

= (2)dx

= + sin-1(x)|01

=

which is approximately equal to
0.79
. Notice that this result
depends critically on the interpretation of the word "average". Here
we are thinking of average of the
y
-coordinate as a function of
x
.
If instead we considered the
y
-coordinate as a function of arc
length around the circle, we would be asking for the average of
sin(t)
on the interval
[0, Π]
, which is
2/Π
.